2. Hydrologic Modeling of Rainfall-runoff Process andIts Application to Real-time Flood Forecasting
Akihiro NAGAI1

The LST model that is physically based conceptual runoff model wasapplied to the hydrological data for 24 years in the Kagami Dam basin(80.8 km2). Close agreement was found between calculated and observed
discharges for flood and long-term runoff. The model was used for realtime flood forecasting combined with updating techniques, such as
backward estimation and the Kalman filter. In order to overcome
nonlinearity of the runoff model, method of statistical linearization wasused in applying the Kalman filter. The real-time flood forecasting system
was verified to perform with good accuracy for 1- and/or 3-hour-ahead
forecasting.
Key Words: flood runoff, real-time flood forecasting, LST model, long-term
runoff, backward estimation, Kalman filter, statistical linearization

1. INTRODUCTION
Flood occurs mainly due to heavy rainfall and is an aspect of the hydrologic cycle.
Considerable research has been carried out to have a better understanding of rainfallrunoff process, and a large number of rainfall-runoff models have been proposed. Inorder to have a reliable model, the process must be modeled properly on the basis of theknowledge of various hydrologic phenomena, such as overland flow, subsurface runoff,
groundwater runoff, infiltration, etc. One of the main potential areas of application of
rainfall-runoff models is in the real-time flood forecasting, flood warning, flood control.
Clearly, the reliability of such forecasts will govern the reliability of such decisions, andthe associated benefits or losses that accrue from them. In this paper, rainfall-runoff
model is applied to analyze flood runoff and long-term runoff successively, and used forreal-time flood forecasting.
2. LONG- AND SHORT-TERM RUNOFF MODEL
The Long- and short-term runoff model (LST model) is one of the physically based
conceptual runoff models. The LST model was proposed to analyze long-term runoff and1

Department of Environmental Management Engineering, Faculty of Environmental

Science and Technology, Okayama University

1

flood (short-term) runoff continuously by Kadoya and Nagai (1988). The model consistsof three storage tanks (Fig.1) and the top tank is divided into two layers. The continuity
equation of each tank is as follows:

dS1 / dt = r − E1 − f − Q1 − Q2 ,dS 2 / dt = f − Q3 − g1

(1)

dS 3 / dt = g 1 − E 2 − Q4 − g 2 ,dS 4 / dt = g 2 − E3 − Q3(2)

where S j is storage, r average depth of rainfall intensity over area, f infiltration
rate, E j evapotranspiration. Q1 is surface runoff, Q2 and Q3 prompt and delayed
subsurface runoff, Q4 and Q5 prompt and delayed groundwater runoff. Runoff Q jand seepage g j are calculated as

Q1 = a1 ( S1 − Z 1 ) m , m = 5 / 3(3)Q2 = a 2 S1 , Q3 = a 3 ( S 2 − Z 3 ) ,g 1 = b2 S 2(4)Q4 = a 4 S 3 , Q5 = a5 S 4 ,g 2 = b3 S 3(5)in which a j and b j are parameters, Z j the height of runoff outlet (Fig.1). Assumingthat Manning's law is applicable for the surface runoff, m = 5 / 3 can be used in (3).
The infiltration rate f from the upper layer to the lower layer of the top tank isf = b1 ( Z 2 + Z 3 − S 2 )(6)When there is enough water in the upper layer, Horton's infiltration equation isexpressed in terms of model parameters. In case of S 2 > Z 3 , the following relations areobtained.

f = f C + ( f 0 − f C ) exp(− kt )f C = [b1b2 ( Z 2 + Z 3 ) + a3b1 Z 2 ] / kk = a 3 + b1 + b2
a1
S1

Ｑ1

(Z1)

a2
b1

Ｑ2

(Z2)

fa3
S2

Ｑ3

(Z3)
b2

g1
S3

a4
Ｑ4
b3

g2
S4

a5
Ｑ5Fig. 1 LST model2(7)(8)(9)

NKagami Dam Basin（80.8 km2）

1000

600800(B)
400(C)

400200800600(D)200400400020Rain Gauge(A)〜(D)
1Kagami Dam(A)0123kmFig. 2 Outline of the Kagami Dam basin
3. H
HYDROLOGICALYDROLOGICAL DATA
The LST model was applied to hydrological data for 24 years from 1975 to 1998 inthe Kagami Dam basin (Fig.2; 80.8 km2) located in the Kochi Prefecture. The data
include 20 floods whose peak discharges vary from 294 to 1,400 m3/s (equivalent to 13.0mm/h to 75.7 mm/h). Annual precipitation and runoff of the basin were 3,107 mm/y and2,357 mm/y respectively on the average for 24 years.
There are 4 telemetering rain gauges in the basin (Fig.2). Rainfall data of eachstation is gathered to the operation room of the dam reservoir. The average depth ofrainfall over the basin area was estimated by the method of arithmetic mean.
For the estimation of areal evapotranspiration, potential evapotranspiration E P
was calculated by Makkink’s equation (Brutsaert, 1982).

EP = aγ
RS + b∆ +γ(10)where RS is global short-wave radiation, as equivalent rate of evaporation, Δ slope ofsaturation water vapor pressure curve, γ

psychrometric constant. The regionalparameters a, b were obtained as a = 0.756 , b = 0.36 mm/d (Nagai, 1993b), with dataat the Kochi meteorological observatory, which is the nearest station from the basin.
For the calculation of (10), the following approximation was made (Kadoya and Nagai,1988).

3

γ1=∆ + γ 1.05 + 1.4 exp(−0.0604T )(11)

where T is temperature (℃).
The areal evapotranspiration ET was estimated with the following rule.

ET = E P ;(daily rainfall)<1mm/d

ET = 0 ;(daily rainfall)≧1mm/d

4. APPLICABILITY OF LST MODEL
The unknown parameters of the LST model were identified with the data of twoyears (1975 and 1976) by Standardized Powell method, which is one of themathematical optimization techniques, under the objective function of (12). The relativeerror defined by (13) is also used for the evaluation of the simulation results.
21 N (QC , j − QO , j )J XS = åN j =1
QO , jJ RE =→ min

1 N | QC , j − QO , j |
× 100
å Q
N j =1O, j(m=5/3)(12)0.0066
(50)
0.036

(13)

0.055

where QO and QC are the observed and calculated
discharges respectively, and N is the sample size.

(84)
0.018
(40)

0.00720.00510.00300.000079

Rainfall(mm/d)

Fig.3 Model parameters (mm-h unit)
050
100
1502001992
10(mm/d)

Runoff Depth

100

10.1
1992/01/01

Observed
Calculated by LST Model
1992/03/01

1992/05/01

1992/07/01

1992/09/01

Fig. 4(a) Simulated daily runoff of 1992

4

1992/11/01Rainfall(mm/d)

0100
2003001993(mm/d)

10Observed
Calculated by LST Model

10.1
1993/01/01

1993/03/01

1993/05/01

1993/07/01

1993/09/01

1993/11/01

Rain (mm/h)

Fig. 4(b) Simulated daily runoff of 1993

010
2030
4050
1000

Kagami Dam Basin

43( Qp=796 m /s)

Discharge (m /s)

23100
4
2

10Observed
Calculated by LST Model

421

15.5
Aug. 15, 1975

16.0
Aug.1616.5

17.0
17.5 Aug.18
18.0
18.5
Aug.1719.0
19.5
Aug.1920.0

Rain (mm/h)

Fig. 5(a) An example of simulated flood runoff

02040Kagami Dam Basin

642(80.8km )23Discharge(m /s)

Runoff Depth

100

100
64Observed
Calculated by LST Model

210023.5
12Sept. 23, 1998

024.0

1224.5025.0

1225.5
Sept.25

Sept.24

0
26.0

26.5
12
Sept.26

Fig. 5(b) An example of simulated flood runoff

5027.0

2

LST Model (Kagami Dam Basin)
20 Floods

3

Calculated Discharge (m /s)

1000

987
6

Verification(1975-1998)

5432100

98765432101023456 7 8 92100

33456 7 8 921000

Observed Discharge (m /s)

Fig.6 Comparison between observed and calculated discharges for all floods
The identified model is shown in Fig.3. When the applicability of the model isevaluated with the relative error, the error for daily runoff was 37.1% on the average for24 years. Examples of the calculated daily runoff hydrograph are shown in Fig. 4(a) andFig. 4(b). As regards flood runoff, the mean relative error was 14.2% for every 1-hour
discharge over 40 m3/s for 20 floods. Examples of calculated hydrograph are shown inFig.5 (a) and Fig. 5 (b), and comparison between observed and calculated discharges for
all floods is shown in Fig. 6. Good results were obtained in the continuous simulation offlood and long-term runoff for 24 years by using the LST model.
5. UPDATING TECHNIQUES FOR REAL-TIME FLOOD FORECASTING
A major problem that often occurs in real-time forecasting is that the forecasted
value is different from the observed value when the latter becomes available. Thisdifference is due to oversimplification of the rainfall-runoff process, or uncertainties ofthe model structure and/or parameters and of the hydrological information, etc. As
shown earlier the identified runoff model has good applicability in the Kagami Dam
basin, the difference is supposed mainly due to the estimation of the areal average
rainfall. The rainfall estimation error is thought to influence the storage depth of the
top tank, and its depth of the upper layer of the top tank was selected as the statevariables for real-time correction. As the storage depth selected as the state variable isalmost control flood runoff, we can expect the accuracy of the forecast to become
improved when the depth is updated adequately.
The state variable was updated by the following two methods. One method isbackward estimation technique that has proposed by Nagai and Yomota (1990). The

6other method is the Kalman filter, the theory of which is for linear system and is foundin the textbook on optimal control (Gelb, 1974). In order to linearize nonlinear function
such as (3) in the LST model, the statistical linearization technique was used. Thetechnique does not require the existence of the derivative of the nonlinear system, andis generally more accurate than the Taylor series expansion method. In the applicationof the statistical linearization, the expectation of nonlinear function was calculated by
Hermite-Gauss integration formula (Abramowitz and Stegun, 1972).
In the backward estimation technique, the storage depth at current time isestimated backward directly from the observed runoff at current time on the bases of
the relationship between storage depth and runoff discharge. For the LST model, direct
runoff QS is estimated from observed runoff QO as

QS = QO − (Q3 + Q4 + Q5 )(14)

The state variable x ≡ S1 is determined in such a way that QS = (Q1 + Q2 ) , that is,x > Z1(15)x ≤ Z1(16)If we define the following function F ( x ) , the value of x is the root of F ( x ) .
QS = a1 ( x − Z1 ) m + a2 x ,QS = a2 x ,F ( x) = a1 ( x − Z1 ) m + a2 x − QS ,x > Z1(17)
F ( x) = a2 x − QS ,x ≤ Z1(18)In case of x > Z 1 , as the value of x is not obtained analytically, the j-th approximation( j)of x , x , is calculated by Newton’s method asx ( j ) = x ( j −1) −where F ' ( x( j −1)

F ( x ( j −1) )F ' ( x ( j −1) )(19)) is the differential coefficient of F ( x) at x = x ( j −1) .

The backward estimation technique has been shown to correspond theoreticallyto the Kalman filter without measurement error (Nagai and Yomota, 1990).
6. RESULTS OF REAL-TIME FLOOD FORECASTING
We carried out real-time flood forecasting with the LST model in combination with
updating techniques. Examples of the flood forecasting are shown in Fig. 7 and Fig. 8.
Comparison between observed and predicted discharges is shown in Fig. 9 for all floods.
Table 1 gives the accuracy of 1- and/or 3- hour-ahead forecasting for 20 floods, in which
the relative error is evaluated for every 1-hour discharge over 40 m3/s. For convenience
future rainfall intensity was assumed to be same as present intensity in cases (A), (B)and (C). In case (D), future rainfall intensity was assumed known. The results aresummarized as follows:(1) Updating techniques improve the accuracy remarkably compared with no filter
case, and therefore the updating technique is indispensable for real-time floodforecasting.
(2) Two updating techniques that are the backward estimation technique and the7Kalman filter, give almost same results.
(3) For more accurate forecasting, more research must be made on the prediction offuture rainfall.
7. CONCLUSIONSThe LST model was used in order to analyze both flood and long-term runoff
successively in the Kagami Dam basin, and the applicability of the model was verified
through the continuous simulation for 24 years. Furthermore, the LST model wasutilized for real-time flood forecasting with two updating techniques. It was shown thatthe 1- and/or 3-hour ahead forecasting of flood runoff was performed with good accuracy.
Future research must be made on the rainfall prediction for more accurate real-timeforecasting.
Table 1 Accuracy of real-time flood forecasting
Case(A)
(B)
(C)
(D)

(No filter)
Backward estimation
Kalman filter(Backward estimation)

Relative error, J RE (%)
1-hour 2-hour
3-hour19.8%
24.3%
34.2%
6.6%
14.2%
24.4%
7.5
14.3
23.5
8.9
11.5
12.5

Future rainfall
Present rainfall
Present rainfall
Present rainfall
Assumed known

Relative error J RE (%) is evaluated for QO ≥ 40m / s.

Rain (mm/h)

3020406080

3Discharge (m /s)

1000

100

Observed
2-hour ahead Forecasting
Backward Estimation
Kalman Filter

101891011121314Sept. 1976

Fig.7 An example of real-time flood forecasting

815Rain (mm/h)

02040
1000
62100
64Observed
2-hour ahead Forecasting( LST Model )Backward Estimation
Kalman Filter

21064
2

1
12:00
1975/08/15

12:00
1975/08/16

12:00
1975/08/17

12:00
1975/08/18

12:00
1975/08/19

Fig.8 An example of real-time flood forecasting

21 hour ahead Forecasting (J RE=7.5%)
1000
3Calculated Discharge (m /s)

3Discharge (m /s)

4

98
76

LST Model
Kalman Filter

54
3

2100

98
7654
32

10102

34

56 7 8 92100

3453

6 7 8 92

1000

Observed Discharge (m /s)

Fig. 9 Comparison between observed and predicted discharges for all floods(1-hour ahead prediction)

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REFERENCES
Abramowitz, M. and Stegun, I. (eds.): Handbook of Mathematical Functions, Dover Pub,1972.
Brutsaert, W. (1982): Evaporation into the Atmosphere, D. Reidel Pub. Com., pp.222223.
Gelb, A. (ed.)(1974): Applied Optimal Estimation，M.I.T. Press，374pp.
Kadoya, M. and Nagi, A. (1988): Development of the Long and Short Terms Runoff
Model, Trans. of Japanese Society of Irrigation, Drainage and Reclamation
Engineering, 136, pp.31-38.
Kadoya, M. and Tanakamaru, H. (1988): Flood Runoff Forecasting with Long and ShortTerms Runoff Model, Proc. of 6th Congress Asian and Pacific Regional DivisionInternational Association for Hydraulic Research (APD-IAHR), pp.183-190.
Nagai, A. and Yomota, A. (1990): Application of Long- and Short-term Runoff Model toForecasting of Flood Discharge in Yoshii River, Trans. of the Japanese Society ofIrrigation, Drainage and Reclamation Engineering, 147, pp.95-102.
Nagai, A. (1993a): Extension of the Tank Model to Continuous Simulation of Long- andShort-term Runoff, Trans. of the Japanese Society of Irrigation, Drainage andReclamation Engineering, 167, pp.89-95.
Nagai, A. (1993b): Estimation of pan evaporation by Makkink equation, Jour. of Japan
Society of Hydrology & Water Resources, 6(3), pp.238-243.

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